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100 = No. of households 55 = No. of MP3s for sure (so assuming that min of 55 households have all 3)

This is incorrect.

How can 100 households have all 3 when only 55 households have MP3 players? 55 is the MAXIMUM number of households that can have all 3, not minimum.and 10 is the minimum number of households that must have all 3 (explained in solutions on first page)So 55 - 10 = 45
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KarishmaVeritas Prep GMAT Instructor

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Re: In a village of 100 households, 75 have at least one DVD player, 80
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11 Dec 2016, 03:52

[quote="VeritasPrepKarishma"]hiredhanak: I am assuming you are looking for a venn diagram solution to this question..

It is pretty simple.First of all maximum number of households: We want to bring the circles to overlap as much as possible.80 - Cell phone75 - DVD55 - MP3Lets take Cell phone and DVD circles since they will have maximum overlap. They must overlap in 55 households so that total number of households is 100. Now put the MP3 households in a way to maximize all three overlap.

Attachment:

Ques1.jpg

So at most 55 households can have all 3.hi karishma

i do not understand. how did you derive at most 55 households can have all 3 by Van diagram

When discussing the maximum overlap case, none NEEDN'T be 0. It may be, it may not be.

Put the three circles within each other. The 75 circle within the 80 circle and the 55 circle within the 75 circle. The overlap will be 55 in that case and none = 20. The figure only shows one of the possible ways of obtaining the maximum.

In the case of minimum, you would want the circles to lie as far apart as possible. If none is anything other than 0, the circles would need to overlap more. Say none = 10, the circles of 80 and 75 would need to have an overlap of 65. So the 55 circle can occupy 25 but an overlap of 30 will be needed. Hence minimum overlap will increase. To minimize the overlap, we will need None = 0.
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Re: In a village of 100 households, 75 have at least one DVD player, 80
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16 Aug 2017, 15:30

I often just don't get the complexity of such kind of questionsmax overlap is 55 (as this is the max number of 3 types of households that can have all three devices)min overlap for 3 devices is 75+55+80 - 100*2 (max number for households that can have 2 devices) = 210 - 200 = 10

Re: In a village of 100 households, 75 have at least one DVD player, 80
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25 Oct 2017, 03:54

Here,the maximum households having three will be 55Now,if we want to get the minimum number of households having three,we need to find at least how many have two of things.So,At least DVD and cell phone=(75+80)-100=55 At least DVD and MP3=(75+55)-100=30 At least cell and MP3 =(80+55)-100=35 now,minimum number=(75+80+55-30-35-55)-100=10 so,the difference is (55-10)=10 Hope it's clear now

In a village of 100 households, 75 have at least one DVD player, 80
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Updated on: 19 Jan 2018, 06:30

Bunuel wrote:

Hussain15 wrote:

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

A. 65

B. 55

C. 45

D. 35

E. 25

For me the best way to solve this problem is not use Venn diagram or formulas but to draw simple bars (note: each dash is 5):

maybe it is very obvious, but I am facing bit troubleplease guide me through my explanation

at least 55 people own MP3 players, so it is very natural to assume that maximum number of people who can have all three is 55okay

now, to have the lowest possible overlapping

100 - 80 = 20 people have no phones (but may have DVDs)100- 75 = 25 people have no DVDs (but may have phones)

so, you have distributed 45 mp3 players to these 45 villagers, causing no overlapping. so far so good

but we are left with 10 more mp3 players to distribute, but one cannot provide these players to the inhabitants who already have the mp3 players because doing so will add up the number of people who own mp3s to 45, a condition that is unwarranted by the question. So, these 10 people must be coming from the groups of DVDs or??? / and ???? phones holders. Here I am stumped, because

these 10 people can have only phones earlier and are now getting mp3sORthese 10 people can have only DVDs earlier and are now getting mp3sin this case there is no overlapping .....

OR

these people can have phones + DVDs earlier and are now getting mp3sin this case there is overlapping....

thanks in advance, man

Originally posted by testcracker on 23 Nov 2017, 10:27.
Last edited by testcracker on 19 Jan 2018, 06:30, edited 1 time in total.

Re: In a village of 100 households, 75 have at least one DVD player, 80
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15 Jan 2018, 09:36

1

VeritasPrepKarishma wrote:

hiredhanak: I am assuming you are looking for a venn diagram solution to this question..

It is pretty simple.First of all maximum number of households: We want to bring the circles to overlap as much as possible.80 - Cell phone75 - DVD55 - MP3Lets take Cell phone and DVD circles since they will have maximum overlap. They must overlap in 55 households so that total number of households is 100. Now put the MP3 households in a way to maximize all three overlap.

Attachment:

Ques1.jpg

So at most 55 households can have all 3.

Now, minimum number of households: We want to take the circles as far apart from each other as possible. Now put the MP3 households in a way to minimize all three overlap. So make the MP3 households occupy the shaded region i.e. region occupied by DVD players alone and cell phone alone. You will be able to adjust 45 MP3s outside the common area but you will need to put 10 of the MP3 households in the common area. So minimum overlap is 10.

Attachment:

Ques2.jpg

x - y = 55 - 10 = 45

Responding to a pm: Explaining the min case.

We want minimum overlap of all three. So we need to spread the 3 around in such a way that all 3 overlap the least. Out of 100 households, 80 have a cell phone. We have 20 households leftover without a cell phone. Now 75 have a DVD so we reduce overlap by putting 20 DVDs in the leftover 20 households. Rest 75 - 20 = 55 will need to be overlapped with DVDs.

So now we have 20 households with just DVDs, 25 with just cell phones and 55 with both.Now we also have to distribute 55 MP3s. Note that overlap of all 3 has to be reduced as much as possible. So we should give minimum to the households that already have both. The rest of the 45 households get MP3s (not these 45 have 2 things each). But we still have 10 MP3s leftover. These will need to be given to the households which have DVDs and cell phones both. So minimum overlap of all 3 is 10.
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Re: In a village of 100 households, 75 have at least one DVD player, 80
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19 Jan 2018, 06:39

gmatcracker2017 wrote:

Bunuel wrote:

Hussain15 wrote:

In a village of 100 households, 75 have at least one DVD player, 80 have at least one cell phone, and 55 have at least one MP3 player. If x and y are respectively the greatest and lowest possible number of households that have all three of these devices, x – y is:

A. 65

B. 55

C. 45

D. 35

E. 25

For me the best way to solve this problem is not use Venn diagram or formulas but to draw simple bars (note: each dash is 5):

maybe it is very obvious, but I am facing bit troubleplease guide me through my explanation

at least 55 people own MP3 players, so it is very natural to assume that maximum number of people who can have all three is 55okay

now, to have the lowest possible overlapping

100 - 80 = 20 people have no phones (but may have DVDs)100- 75 = 25 people have no DVDs (but may have phones)

so, you have distributed 45 mp3 players to these 45 villagers, causing no overlapping. so far so good

but we are left with 10 more mp3 players to distribute, but one cannot provide these players to the inhabitants who already have the mp3 players because doing so will add up the number of people who own mp3s to 45, a condition that is unwarranted by the question. So, these 10 people must be coming from the groups of DVDs or??? / and ???? phones holders. Here I am stumped, because

these 10 people can have only phones earlier and are now getting mp3sORthese 10 people can have only DVDs earlier and are now getting mp3sin this case there is no overlapping .....

OR

these people can have phones + DVDs earlier and are now getting mp3sin this case there is overlapping....

thanks in advance, man

yes, I think I have got the issue

it is possible that these 10 people can have all three items, so minimum overlapping is 10

Re: In a village of 100 households, 75 have at least one DVD player, 80
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08 Feb 2018, 00:23

make it more simple. using ven diagram for 2 factors, dvd and cellphone first.

look at two factors, person with dvd is 75 and with cellphone is 80.so, there are 55 person with at least both dvd and cellphone because there is only total 100. this overlapped area contain 55 persons. simple

now there is 55 person with mp3. to maximize , we put all of 55 persons with mp3 in the the overlapped area between two groups above. perfect. there are maximum 55 person with dvd,cellphone and mp3

to minimize, look at the overlapped group of 55 person with dvd and cell phone. among 55 person with mp3, we put the minimum number in the the overlapped area of the two groups.

LOOK AT area of 75 person with dvd. we can put maximum 25 persons with mp3 into group of 80 person with cellphone because we can not put more. if we put 26 there would be 75+26=101 persons. WRONG.similarly,,look at area of 80 persons with cellphone, we can put maximum 20 person with mp3 into group of 75 persons with dvd. if we put 21, there would be 80+21= 101 person, impossible.

so, from 55, we put 25 and 20 into two above group, the number left is 10.

mimimum is 10

this is hard. but can be learned. if we can not do this, we can still get 49 on math. dont worry

Re: In a village of 100 households, 75 have at least one DVD player, 80
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10 Jan 2019, 11:43

PiyushK wrote:

My imagination about minimum overlapping percentage.

Attachment:

The attachment overlapp.jpg is no longer available

After spending all day obsessing over this problem your diagram inspired me to find a solution that works for me.

We know the formula for three overlapping sets is A+B+C-(exactly two overlap) -2*(exactly three overlap) = Total

So A+B+C-(exactly two overlap)-2*(exactly three overlap)=100. Now when we add up A,B,C we get 75+55+80=210, and subtracting 100, we have 110 extra households that need to be distributed among the three. We can only distribute to A,B,C until their total equals 100 as shown in the image I have attached (since there are max 100 holds. This means we can distribute 20, 25, and 45 = 90 items total. Subtracting that from our 110 extra households, and we get 20. Remember, that since from the formula we count the three overlap twice, the minimum 3 overlap area will be 20/2=10. Now, we know the max overlap is the smallest of the three groups = 55, therefore our answer is X-Y = 55-10=45 D

Attachments

Untitled.png [ 29.85 KiB | Viewed 31 times ]

gmatclubot

Re: In a village of 100 households, 75 have at least one DVD player, 80 &nbs
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10 Jan 2019, 11:43